| L(s) = 1 | + 4.86·2-s − 9·3-s − 8.29·4-s + 66.7·5-s − 43.8·6-s − 185.·7-s − 196.·8-s + 81·9-s + 324.·10-s − 563.·11-s + 74.6·12-s + 767.·13-s − 902.·14-s − 600.·15-s − 689.·16-s − 1.73e3·17-s + 394.·18-s − 2.28e3·19-s − 553.·20-s + 1.66e3·21-s − 2.74e3·22-s − 529·23-s + 1.76e3·24-s + 1.32e3·25-s + 3.73e3·26-s − 729·27-s + 1.53e3·28-s + ⋯ |
| L(s) = 1 | + 0.860·2-s − 0.577·3-s − 0.259·4-s + 1.19·5-s − 0.496·6-s − 1.42·7-s − 1.08·8-s + 0.333·9-s + 1.02·10-s − 1.40·11-s + 0.149·12-s + 1.25·13-s − 1.23·14-s − 0.689·15-s − 0.673·16-s − 1.45·17-s + 0.286·18-s − 1.45·19-s − 0.309·20-s + 0.825·21-s − 1.20·22-s − 0.208·23-s + 0.625·24-s + 0.425·25-s + 1.08·26-s − 0.192·27-s + 0.370·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 - 4.86T + 32T^{2} \) |
| 5 | \( 1 - 66.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 185.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 563.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 767.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.73e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.28e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 3.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.49e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.62e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.05e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.33e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.10e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.86e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.26e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.75e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.71e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.06340480865057494440260101337, −12.79431758802094501570951388330, −10.90603518019873401460696836829, −9.897593709722055347303419539574, −8.759210367298496568181777959885, −6.33767313296198598559639394380, −5.95535556812359418736758545179, −4.41612425300882700498927474010, −2.67331697353826190759962331456, 0,
2.67331697353826190759962331456, 4.41612425300882700498927474010, 5.95535556812359418736758545179, 6.33767313296198598559639394380, 8.759210367298496568181777959885, 9.897593709722055347303419539574, 10.90603518019873401460696836829, 12.79431758802094501570951388330, 13.06340480865057494440260101337
